Question: Express the quotient $1121_5 \div 12_5$ in base $5$.
Long division in base $5$ follows the same format as that in base $10$.

\[
\begin{array}{c|cccc}
\multicolumn{2}{r}{} & & 4 & 3 \\
\cline{2-5}
12 & 1 & 1 & 2 & 1 \\
\multicolumn{2}{r}{1} & 0 & 3 & \downarrow \\ \cline{2-4}
\multicolumn{2}{r}{} & & 4 & 1 \\
\multicolumn{2}{r}{} & & 4 & 1 \\ \cline{4-5}
\multicolumn{2}{r}{} & & & 0
\end{array} \]Giving us a final answer of $\boxed{43_5.}$